Fundamentally, the RPGA is WotC's Organized Play group for role-playing games (analagous to the role that DCI plays for Magic: the Gathering). It exists to promote play of RPGs, specifically WotC's RPGs and others that use the D20 system.
They do this through creating and adminstering a number of different RPG campaigns that you can play in.
RPGA directly oversees four different ongoing campaigns:
- Living Greyhawk
- Legacy of the Green Regent (set in Forgotten Realms, but being wrapped up right now)
- Mark of Heroes (set in Eberron)
- Living Force (Star Wars, also nearing its conclusion)
Each of these campaigns release new adventures on a regular basis (typically a dozen or more a year); those adventures can be played at conventions, game days (like at your FLGS), or even with your home group. At many big RPG conventions, you'll see a significant RPGA presence; at a con like GenCon or Origins, there'll be hundreds of tables of RPGA games played.
In "Living" campaigns, you create a PC at 1st level, just like in your home game. You can then play that PC in any adventure for that campaign, anywhere in the world that you can find a game. So, if you create a Living Greyhawk character to play at your FLGS, you can play that same character in Living Greyhawk adventures at GenCon, or at home.
Green Regent and Mark of Heroes work a little differently. They're called "D&D Campaigns", and one of the biggest differences is that every PC gets "advanced" to a higher level on a regular basis in these. Thus, at any given time, all the PCs in Mark of Heroes are going to be roughly the same level (I think that's what Crothian was referring to by "gain levels at the same time"...and that only happens in these two campaigns. In Living campaigns, you generally only gain new levels for your PC through play and XP awards, and you'll see a wide range of different-leveled PCs.)
RPGA also serves as the distribution point for several other "Living" campaigns that are developed by either (a) other publishing companies, using D20, or (b) a private group, in the case of Living Death:
- Living Arcanis (D&D, set in Paradigm Concepts' Arcanis)
- Living Kingdoms of Kalamar (D&D, set in Kenzer & Co.'s Kalamar)
- Living Death (variant D&D, set in a revised version of the old Ravenloft "Masque of the Red Death" 1890s gothic horror setting)
- Living Spycraft (modern espionage, set in AEG's Spycraft setting)
As Crothian mentioned, all of these campaigns have more restrictive rules on character generation and advancement than you'd likely have in your home campaign. Also, the RPGA insists that, when you run an RPGA adventure, you (as the DM) run it "as written", and don't change encounters, modify rewards, etc. All of this helps to make sure that the PCs can be portable, and that the game experience is reasonably consistent.
The RPGA used to charge a yearly membership fee (and that used to cover a subscription to Polyhedron magazine); membership has been free for several years now. If you want to play, all you need is a membership number (which you can get when you attend your first RPGA event). If you want to set up your own events (like running stuff for your home group), you need to pass the "Herald-Level GM" test, which is also free.
William Ronald pointed you to the RPGA's main page; you may also find their
Newcomer's Page useful.
Hope this helps!